Equivalent Multiplication Expressions

10 min

Narrative

The purpose of this How Many Do You See? is for students to use grouping strategies to describe the images they see. Students’ descriptions are recorded, using equations and expressions to support the goal of creating equivalent expressions. The Activity Synthesis encourages students to think about why two expressions can represent the same amount.

Launch

  • Groups of 2
  • “How many thirds do you see? How do you see them?”
  • Display image.
  • 1 minute: quiet think time
     
Teacher Instructions
  • Display image. 
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Record responses.

Student Task

How many thirds do you see? How do you see them?

8 diagrams of equal length. 3 equal parts. 1 part shaded. Total length, 1.

Sample Response

  • 8: 8 groups of 13\frac{1}{3} or 8×138 \times \frac{1}{3}
  • 4: 4 groups of 23\frac{2}{3} or 4×234 \times \frac{2}{3}
  • 2: 2 groups of 43\frac{4}{3} or 2×432 \times \frac{4}{3}
Activity Synthesis (Teacher Notes)
  • For each way that students see thirds, ask: “What expression should we use to represent the groups of thirds that _____ saw?”
  • If no students suggest “4 groups of 23\frac{2}{3},” ask them how that might be visible in the diagram. (By combining 2 of the thirds from each strip, we can make 4 groups of 23\frac{2}{3}.)
  • Write 8×13=4×238 \times \frac{1}{3} =4 \times \frac{2}{3}, and ask students if they agree or disagree with the statement.
Standards
Addressing
  • 4.NF.4.b·Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. <em>For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)</em>
  • 4.NF.B.4.b·Understand a multiple of <span class="math">\(a/b\)</span> as a multiple of <span class="math">\(1/b\)</span>, and use this understanding to multiply a fraction by a whole number. <span>For example, use a visual fraction model to express <span class="math">\(3 \times (2/5)\)</span> as <span class="math">\(6 \times (1/5)\)</span>, recognizing this product as <span class="math">\(6/5\)</span>. (In general, <span class="math">\(n \times (a/b) = (n \times a)/b.\)</span>)</span>

15 min

20 min